Computational Solid State Physics

計算物性学特論　第７回

7. 　 Many-body effect IHartree approximation, Hartree-Fo

ck approximation andDensity functional method

Hartree approximation

jiji

N

ii

ijji

N

iii

rrvrh

r

erV

mH

),()(

1

4)(

2

1

0

2

1

2

),()()(),,( 22111 NNN rrrrr

)(ri

N-electron Hamiltonian

・ N-electron wave function

i-th spin-orbit

ijji rrdr )()(* ortho-normal set

Expectation value of the energy

drHH *

H

E

1*| dr

pqrrvpqprhpH

EN

qp

N

p

)(2

1)( 2,1

1,1

single electron energy Hartree 　 interaction

Charge density

2

1

212121

1

|)(|

),,()(),,(*)(

)()(

r

drdrdrrrrrrrrrn

rrr

p

N

p

NNN

N

ii

2122112,11,

)(),()(2

1)(

2

1drdrrnrrvrnpqrrvpq

N

qp

Hartree interaction

: charge density

:charge density operator

Hartree calculation for N>>1

ijji rrdr )()(*

)()(])()()([ 111 rrdrrrvrnrh iii

Energy minimization with condition

Self-consistent Schröedinger equation for the i-th state

Electrostatic potential energy caused by

electron-electron Coulomb interaction

N

pp rrn

1

2|)(|)( charge density

Hartree-Fock approximation

Pauli principle Identical particlesSlater determinantExchange interactionHartree-Fock-Roothaan’s equation

jiji

N

ii

ijji

N

iii

rrvrh

r

erV

mH

),()(

1

4)(

2

1

0

2

1

2

：),(

:)(

ji

i

rrv

rh

Many electron Hamiltonian

single electron Hamiltonian

electron-electron Coulomb interaction

Slater determinant

)()()(

)()()(

)()()(

!

1),,(

21

22212

12111

1

NNNN

N

N

N

rrr

rrr

rrr

Nrr

)( ir

),()()()1(!

1),,(

!

12111 NN

N

N rrrN

rr

N

N

21

21

Permutation of N numbers

John Slater

2/))2()1()2()1()(()( 2111 rr

)2()1(2/))()()()(( 21122211 rrrror

spin orbit

N-electron wave function

Properties of Slater determinant

0)( 1 Nrr)()( rr ji

)()( 11 NijNji rrrrrrrr

Pauli principle

Identical Fermi particles

ji rr or

The Slater determinant satisfies both requirements of Pauli principle and identical Fermi particles on N-electron wave function.

If

Ground state energy

H

E

)()()()1(!

1),,(

!

12111 NN

N

N rrrN

rr

1

)()()(

)(*)(*)(*)1(!

1

'1'21'1

!

',12111

'

NN

NN

N

N

rrr

rrrdrdrN

ijji rrdr )()(*

N

N

21

21

Permutation of N numbers

Orthonormal set

Expectation value of Hamiltonian

N

pipiipi

NNi

NN

N

Ni

rrhrdrN

rrrrh

rrrdrdrN

rh

1

'1'21'1

!

',12111

'

)()()(*1

)()()()(

)(*)(*)(*)1(!

1)(

ji

ji

N

ii rrvrhH ),()(

1

N

p

N

p

N

jpp

N

ii prhprrhrdr

Nrh

11 11

)()()()(*1

)(

Expectation value of Hamiltonian

)]()(),()(*)(*

)()(),()(*)(*[)1(

1),(

1,

iqjpjijqip

jq

N

qpipjijqipjiji

rrrrvrr

rrrrvrrdrdrNN

rrv

)]()(),()(*)(*

)()(),()(*)(*[2

1),(

1,

iqjpjijqip

jq

N

qpipjijqipji

jiji

rrrrvrr

rrrrvrrdrdrrrv

Expectation value of many-electron Hamiltonian

])()([2

1

)(

2,12,11,

1

qprrvpqpqrrvpq

prhpH

E

N

qp

N

p

Coulomb integral

Exchange integralHartree term: between like spin electrons and between unlike spin electrons

Fock term: between like spin electrons

Exchange interaction

suppression of electron-electron Coulomb energy

No suppression of electron-electron Coulomb energy

Ｘ

p p qq

no transfer transfer

gain of exchange

energy

No exchange

energy

Pauli principle

Hartree-Fock calculation (1)

m

jiijjkik

N

k

SCC1,1

*|

jiijS

),,1(1

NiCm

jjjii

jExpansion by base functions

Hartree-Fock calculation (2)

m

jijpipij CChprhp

1,

*|)(|

klrrvijCCCCpqrrvpq lqkpjq

m

lkjiip )()( 2,1

*

1,,,

*2,1

N

qlqjqjl CCP

1

*

klrrvijPCCpqrrvpq jlkp

m

kiip

m

lj

N

q

)()( 2,1*

1,

*

1,2,1

1

jiij rhh |)(|

Calculation of the expectation value

Hartree-Fock calculation (3)

N

p

m

kijlkpip

m

lj

N

p

m

jijpipij

klrrvijPCC

CChH

1 1,21

1,

1 1,

|),(|*2

1

*||

N

qlqjqjl CCP

1

*

Expectation value of N-electron Hamiltonian

Hartree-Fock calculation (4)

),,1,,,,1(,01

mjiNkCSEF jk

m

jijkij

m

jiklijij ljvikjlvikPhF

1,

0*

ikC

EMinimization of E with condition

N

qlqkqkl CCP

1

*

ijji rrdr )()(*

Hartree-Fock-Roothaan’s equation

Exchange interaction is also considered in addition to electrostatic interaction.

Hartree-Fock calculation (5)

01

jk

m

jijkij CSEF

m

jiklijij

jkijij

ljvikjlvikPhF

CSF

1,

,,, CSFESCFC

N

qlqkqkl CCP

1

*

Self-consistent solution on C and P

m: number of base functions

N: number of electrons

Schröedinger equation for k-th state

Density functional theory

Density functional method to calculate the ground state of many electrons

Kohn-Sham equations to calculate the single particle state

Flow chart of solving Kohn-Sham equation

Many-electron Hamiltonian

)(rvVTH extee

T: kinetic energy operator

Vee: electron-electron Coulomb interaction

vext: external potential

Variational principles

Variational principle on the ground state energy functional E[n]: The ground state energy EGS is the lowest limit of E[n].

Representability of the ground state energy.

n

een

GS

GSGSextGS

VTnF

nFrnrrvdE

minmin

3

||][

][)()(

'''|),'',',(|)( 2 dxdxdxxxNrn n :charge density

Density-functional theory

Kohn-Sham total-energy functional for a set of doubly occupied electronic states

2)(2)(

iin rr

})({)]([)()(

2

)()(2

2}][{

33

2

332

2

*

IionXC

ionii

ii

EnEnne

nVm

E

Rrrrddrr

rr

rdrrrd

Hartree term Exchange correlation term

Kohn-Sham equations

)()()()()(2

22

rrrrr iiiXCHion VVVm

rdrr

rr

32 )()(

neVH

)(

)]([)(

r

rr

n

nEV XC

XC

)]([ rnEXC

: Hartree potential of the electron charge density

: exchange-correlation potential

: excahnge-correlation functional

Kohn-Sham eigenvalues

　i

iifn2

)()( rr

i

iiiis mffnT

2]),([

2r

ii f

E

1

0101 )( dffEE ifNfNii

Janak’s theorem:

: Kinetic energy functional

If f dependence of εi is small, εi 　 means an ionization energy.

Local density approximation

)]([)( hom rr nxcxc

),( rr XCn

)(hom nxc

rr

rrrr

),()(

2)]([ 33

2xc

xc

nnrdrd

enE

1),(3 rrXCnrdＳｕｍ Ｒｕｌｅ :

: exchange-correlation hole

rdrrr 3)())(()]([ nnnE XCXC

: Exchange-correlation energy per electron in homogeneous electron gas

Local-density approximation satisfies the sum rule.

exchange hole distribution for like spin

nX(r12)

1),(3 rrXnrd

Bloch’s theorem for periodic system

)(]exp[)( rrkr ii fi

G

GrGr ]exp[)(

,icf

ii

G

Gk rGkr )(exp)( , icii

)()( rariiff

G : Reciprocal lattice vector a : Lattice vector

Plane wave representation of Kohn-Sham equations

GkGk

GGG

GGGG

GGGk

,,

22

)()(

)(2

iiiXCH

ion

ccVV

Vm

Supercell geometry

Point defect Surface Molecule

Conjugate gradient method

Molecular-dynamics method

Flow chart describing the computational procedure for the total energy calculation

Hellman-Feynman force on ions (1)

: for eigenfunctions

occupi

i

i

i

ii d

dE

d

dErd

EE

d

d

:

*

*3 )(

)(

)(

)(]},[{

r

r

r

r

)()(*

rr i

i

HE

occupi

iii

ii Hd

d

d

dH

EE

d

d

:

]},[{

0 iiiiii

i d

dH

d

d

d

dH

})({)]([)()(

2

)()(2

2}][{

33

2

332

2

*

IionXC

ionii

ii

EnEnne

nVm

E

Rrrrddrr

rr

rdrrrd

n

nion

n

ion

n R

RrVrdrn

R

EF

)()(

Electrostatic force between an ion and electron charge density

Electrostatic force between ions

Hellman-Feynman force on ions (2)

Problems 7

Derive the single-electron Schröedinger equations in Hartree approximation.

Derive the single-electron Schröedinger equations in Hartree-Fock approximation.

Derive the Kohn-Sham equation in density functional method.

Solve the sub-band structure at the interface of the GaAs active channel in a HEMT structure in Hartree approximation.